The ${\mathrm{\ell }}^p$-metrization of functors with finite supports

Let $p\in [1,\mathrm{\infty }]$ and $F:\mathbf{S}\mathbf{e}\mathbf{t}\to \mathbf{S}\mathbf{e}\mathbf{t}$ be a functor with finite supports in the category $\mathbf{S}\mathbf{e}\mathbf{t}$ of sets. Given a non-empty metric space $(X,d_X)$, we introduce the distance $d_{FX}^p$ on the functor-space $FX$ as the largest distance such that for every $n\in \mathbb{N}$ and $a\in Fn$ the map $X^n\to FX$, $f\mapsto Ff(a)$, is non-expanding with respect to the ${\mathrm{\ell }}^p$-metric $d_{X^n}^p$ on $X^n$. We prove that the distance $d_{FX}^p$ is a pseudometric if and only if the functor $F$ preserves singletons; $d_{FX}^p$ is a metric if $F$ preserves singletons and one of the following conditions holds: (1) the metric space $(X,d_X)$ is Lipschitz disconnected, (2) $p=1$, (3) the functor $F$ has finite degree, (4) $F$ preserves supports. We prove that for any Lipschitz map $f:(X,d_X)\to (Y,d_Y)$ between metric spaces the map $Ff:(FX,d_{FX}^p)\to (FY,d_{FY}^p)$ is Lipschitz with Lipschitz constant $\mathrm{L}\mathrm{i}\mathrm{p}(Ff)\le \mathrm{L}\mathrm{i}\mathrm{p}(f)$. If the functor $F$ is finitary, has finite degree (and preserves supports), then $F$ preserves uniformly continuous function, coarse functions, coarse equivalences, asymptotically Lipschitz functions, quasi-isometries (and continuous functions). For many dimension functions we prove the formula $\dim F^pX\le$\mathrm{d}\mathrm{e}\mathrm{g}$(F)\cdot\dim X$. Using injective envelopes, we introduce a modification ${\check{d}}_{FX}^p$ of the distance $d_{FX}^p$ and prove that the functor ${\check{F}}^p:\mathbf{D}\mathbf{i}\mathbf{s}\mathbf{t}\to \mathbf{D}\mathbf{i}\mathbf{s}\mathbf{t}$, $\check F^p:(X,d_X)\mapsto (FX,\check d^p_{FX})$,inthecategory$$\mathbf{D}\mathbf{i}\mathbf{s}\mathbf{t}$$ of distance spaces preserves Lipschitz maps and isometries between metric spaces.